I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion, but I have not come across it in fields like biology, economics, etc. Are there examples of non-trivial (co)homology appearing outside of pure mathematics?

I think Hatcher has a couple illustrations of homology in his textbook involving electric circuits. This is the type of thing I'm looking for, but it still feels like topology since it is about closed loops. Since the relation $d^2=0$ seems so simple to state, I would imagine this setup to be ubiquitous. Is it? And if not, why is it so special to topology and related fields?

Ghrist has written a number of papers applying homology to different applied fields. See for example his paper on Sensor Networks.
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Jim ConantMar 30 '11 at 19:30

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This question should be Community Wiki, since it's looking for a list of examples rather than an answer to a specific question. I also feel like the question isn't very clear, since any answer has to be homological in nature so fundamentally mathematical. So the question seems somewhat conflicted.
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Ryan BudneyMar 30 '11 at 19:35

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Do you count string theory as "outside the realm of pure mathematics"?
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QfwfqMar 30 '11 at 20:04

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The first paragraph seems to suggest that the question is not so much about homology/topology in the large, but more about the idea of chain complexes. Is my reading correct?
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Yemon ChoiMar 30 '11 at 20:27

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Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: http://www.math.upenn.edu/~ghrist/index.html ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We can form a complex of these sensors and hence its nerve, and use homology to determine whether there are any gaps in the sensor-collection. I've met with him in person and he expressed confidence that this is going to be a big thing of the future.

There are also applications of cohomology to Crystallography (see Howard Hiller) and Quasicrystals in physics (see Benji Fisher and David Rabson). In particular, it uses cohomology in connection with Fourier space to reformulate the language of quasicrystals/physics in terms of cohomology... Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of nonzero homology classes.

Another application is on fermion lattices (http://arxiv.org/abs/0804.0174v2), using homology combinatorially. We want to see how fermions can align themselves in a lattice, noting that by the Pauli Exclusion principle we cannot put a bunch of fermions next to each other. Homology is defined on the patterns of fermion-distributions.

Robert Ghrist is coming to Edinburgh for the Science Festival this year to talk about the Mathematics of holes. (Alas I will be away.) If anyone is around, it will be worth your while to attend!
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José Figueroa-O'FarrillMar 30 '11 at 20:59

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In general, I think that homology will play a role in the mathematics of information. We had a talk by Gunnar Carlsson recently in Edinburgh about "persistent homology" and it was quite an eye opener. See comptop.stanford.edu for instance.
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José Figueroa-O'FarrillMar 30 '11 at 21:00

Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count.

5+7 = 1 2

4 + 5 = 0 9

2 + 8 = 1 0

What is the function on which sends a pair (a,b) to the $0$ or $1$ depending result is greater than 9 or not ?
( e.g. f(5,7)= 1, f(4,5) = 0, f(2,8)= 1).

This is actually a 2-cocycle for group $Z/nZ$ with values in $Z$.

It can be checked directly or...

Let us look on it more conceptually. Consider the standard short exact sequence of abelian groups $0 \to Z \to Z \to Z/n \to 0$. (First map is multiplication by $n$, the second is factorization and will be denoted by $p$).

Choose section $s: Z/nZ \to Z$ (i.e. any map such $ps=Id$, where $p: Z \to Z/nZ$, it is like connection in differential geometry (can be made precise)).

Define $f(a,b)= s(a)+s(b) - s(a+b)$

Note that: a) this function $f(a,b)$ is exactly we talked above

b) from general theory this is 2-cocyle, (it corresponds to this extension, (it it is like "curvature" of connection is differential geomety (can be made precise)).

That is all: we explained why it is group cocycle and what its role.

I would like to learn this 20 years ago when I learned group cohomology as undergraduate, but I learned this 1 ago, doing some engineering work in wireless communication...
I am still surprised that it is not written on the first page of any textbook which deals
with group cohomology, when I am explaining this to my friends most did not know this also and after knowing share my feeling of surprise.

While not on the first page of a textbook, this is written up in an article in the American Mathematical Monthly (Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly (109), no. 9 (2002), p. 796--805)
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Christopher DrupieskiJun 11 '12 at 20:44

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is it easy to generalize to larger digits? how about for multiplication?
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TurboJun 18 '12 at 16:34

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@36min In arxiv.org/abs/hep-th/0212195 we have discussed such a look on connections (actually for more general setup of Courant algebroids), it might be possible to get idea from there, but may be it is not good starting point. I am sorry I do not know reference for standard exposition of this approach in the case of usual connections.
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Alexander ChervovNov 21 '12 at 6:16

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@36min Let me give an idea consider module V over M. Consider the following exact triple: End(V) ->A(V) -> Der(M), where A(M) - the set of all "derivations" of module V. The idea is that connection is exactly the same is a section s: A(V) <- Der(M) !!! The curvature is the following - take two elements a,b in Der(M) , consider F(a,b) = [s(a),s(b)] -s([a,b]) observe that F(a,b) lies actually in End(V) \subset A(V) so we get map F(a,b) : Der(M)^2 -> End(V) - this is curvature
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Alexander ChervovNov 21 '12 at 6:24

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@36min , what is Der(M) - set of all derivations of algebra of functions on M, i.e. all vector fields. What is A(V) - set of all derivations of module V. Map d: V-> V is called derivation of module V, if there exists derivation dd \in Der(M) such that for any element "a" of algebra M , it is true that d(a v) = dd(a) v + a dd(v) !!!!!!!! It is simple, may be I am explaining not in a right way. Is it clear ?
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Alexander ChervovNov 21 '12 at 6:28

Quantum field theory is outside the realm of pure mathematics, makes contact with the real world and features chain complexes and cohomology.

The current paradigm for gauge theories such as the standard model is based on Yang-Mills theories coupled to matter. The quantisation of nonabelian (and, depending on your choice of gauge fixing function, also abelian) Yang-Mills theories features a cohomology theory known by the moniker of BRST, after the inventors: Becchi, Rouet, Stora and, independently, Tyutin. The cleanest proofs of the renormalizability of Yang-Mills theories are cohomological in nature.

My understanding, from conversations with Raoul Bott, is that his early work on electrical circuits and the Bott-Duffin theorem can be intepreted as exhibiting close connections between de Rham cohomology and the laws of electrical circuits, and that this is part of what led him into pure mathematics early in his career.

The mass of a classical mechanical system is an element in the (one-dimensional) second cohomology group of the Lie algebra of the Galilei group. See J. M. Souriau, Stucture des Systèmes Dynamiques, Chap. III, section (12.136). Or in english translation, search inside here for "total mass".

Recently, it was realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.

The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order
in higher dimensions may need higher category to describe them.

The quantum phases with short-range entanglements that break the symmetry
are the familar Landau symmetry breaking states, which can be described by group theory.

So, to understand the symmetry breaking states, physicists have been
forced to learn group theory.
It looks like to understand patterns of many-body entanglements
that correspond to topological order and SPT order, physicists will be forced to
learn tensor category theory and group cohomology theory.
In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.

Anders Björner and László Lovász used bounds on the Betti numbers for the complement of a real subspace arrangement called the $k$-equal arrangement to give a complexity theory lower bound that agreed, up to a scalar multiple, with the previously known upper bound in:

The basic question addressed in their paper (along with other questions of a similar flavor) is how many pairwise comparisons of coordinates are needed to decide if a vector in ${\bf R}^n$ has $k$ coordinates all equal to each other for fixed $k$ and $n$. They observed that this is equivalent to deciding whether the vector lies on the so-called $k$-equal arrangement or in its complement, where the $k$-equal arrangement is the subspace arrangement comprised of the ${n\choose k}$ subspaces where $k$ coordinates are set equal to each other.

To this end, they gave a lower bound on the number of leaves in a linear decision tree -- a tree where one starts at the root, and each time one does a comparison of two coordinates $a_i$ and $a_j$, then one proceeds down to either the $a_i < a_j$ child or the $a_i = a_j$ child or the $a_i > a_j$ child. One reaches a leaf when no further queries are necessary to make a decision as to containment in the arrangement or its complement. The log base 3 of the number of leaves is a lower bound on the depth of the tree, i.e. on the number of queries needed in the worst case.

To get some intuition for why this bound depended fundamentally on the Betti numbers of the complement, consider the $k=2$ case -- where the number of connected components of the complement of the subspace arrangement (which in this case is a hyperplane arrangement) is an obvious lower bound on the number of leaves in any linear decision tree.

Something resembling de Rham complex with differential-algebraic flavor appears in (variant of) control theory, see, for example, G. Conte, C.H. Moog, A.M. Perdon, Algebraic Methods for Nonlinear Control Systems, 2nd ed., Springer, 2006. But, as far as I can tell, they do not use the word "cohomology" explicitly.

Spencer cohomology (which is, essentially, a Lie algebra cohomology) appears as obstructions to integrability of some differential-geometric structures (G-structures) and, through it, of (some) differential equations. Potentially this opens a wide possibilities for applications, and indeed, Dimitry Leites advocates this approach in (some of) his writings. An emblematic publication which is available, unfortunately, only in Russian, is: "Application of cohomology of Lie algebras in national economy", Seminar "Globus", Independent Univ. of Moscow, Vol. 2, 2005, 82-102. The Russian original for "national economy" in the title is (a somewhat pejorative and untranslatable term) "narodnoe khozyai'stvo".

The Aharonov–Bohm effect. Classically, you can't distinguish two electromagnetic potentials which are in the same cohomology class. From quantum viewpoint, they can be distinguished, because an electron changes its phase under parallel transport defined by the connection associated to a potential.

Wheeler's geometrodynamics program contained a subprogram named "charge without charge", which aimed to express the electric charge in terms of geometric and/or topological properties. A wormhole allows the existence of an electromagnetic field without source - hence the name "charge without charge". The two ends of the wormholes behave as particles of opposite electric charge. And all this can be obtained as a solution to Einstein-Maxwell equations. Roots of the approach of Misner and Wheeler can be found in the paper of Einstein and Rosen, and a series of papers of G. Y. Rainich from 1924-1925.

There are some applications of topology/cohomology to combinatorics and combinatoric geometry. One of the earliest examples is surely Lovasz's proof of a bound for the chromatic number of the Kneser graph; he uses the Borsuk-Ulam theorem, which is usually proved by homological methods. A modern exposition can be found here.

Another example is Tveberg's theorem with all its variants on the configuration of points in space (the best results can be found in a recent paper of Blagojevic, Matschke and Ziegler. There are many other results in convex geometry/polytope theory which use topological methods and, in particular, cohomology.